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Question Number 123460 by Dwaipayan Shikari last updated on 25/Nov/20

$${sinh}^{-1}\left(1\right)+{sinh}^{-1}\left(\frac{1}{2^2}\right)+{sinh}^{-1}\left(\frac{1}{3^2}\right)+{sinh}^{-1}\left(\frac{1}{4^2}\right)+....$$$$$$

Commented by Dwaipayan Shikari last updated on 25/Nov/20

$${sinh}^{-1}x=t$$$$\Rightarrow x=sinht\Rightarrow x=\frac{e^t-e^{-t}}{2}\Rightarrow e^t=x\pm \sqrt{x^2+1}\Rightarrow t=log(x\pm \sqrt{x^2+1})$$$${sinh}^{-1}x=log(x\pm \sqrt{x^2+1})$$$${sinh}^{-1}\left(1\right)+{sinh}^{-1}\left(\frac{1}{2^2}\right)+{sinh}^{-1}\left(\frac{1}{3^2}\right)+...=log((1+\sqrt{2})\left(\frac{1+\sqrt{17}}{4}\right)\left(\frac{1+\sqrt{82}}{9}\right)...)$$

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